Common Best Proximity Coincidence Point Theorem for Dominating Proximal Generalized Geraghty in Complete Metric Spaces
In this paper, we introduce a new concept of dominating proximal generalized Geraghty for two mappings and prove the existence and uniqueness of a common best proximity coincidence point in complete metric spaces. And also, we give an example for the main theorems. The main theorem is a generalization and improvement of some well-known theorems.
The best proximity point problems have been attracted to many researchers as there are various applications in real-world problems. The optimization problem is one of the applications that benefit from the best proximity point theory. In other words, it helps finding an approximate solution to the fixed point problems even the mapping itself does not have a fixed point (see [1–23]). In literature, most works focus on suggesting suitable conditions to promise the existence of approximate optimal solutions. These results give the best proximity point theorem in a variety of approaches.
For instance, the work of Geraghty  is one of several important results inspired by the Banach contraction principle for the existence of fixed points for self mappings in metric spaces. In fact, this result generalizes previous concepts by introducing the class of all mappings such that
In 2012, Basha  proposed a result on common best proximity points with a property called proximal commutativity of mappings. Later, Kumam and Mongkolekeha  considered common best proximity point theorems for proximity commuting mappings. In addition, this study has been done according to Geraghty’s work in complete metric spaces. After that, Chen  established the definition of a mapping generally dominates a mapping and accomplished theorems of existence and uniqueness of common best proximity points for a pair of nonself mappings. Lately, Ayari  improved the class of Geraghty and defined a new class of the mappings such that
Accordingly, the existence and uniqueness of best proximity points is guaranteed for -proximal Geraghty nonself mappings on a closed subset of a complete metric space.
To generalize previous results, we are interested to extend our study to common best proximity coincidence points for two mappings under certain conditions. Specifically, we investigate the existence and uniqueness of common best proximity coincidence points for any pairs of two mappings that are dominating proximal generalized Geraghty on a complete metric space. In particular, this work is organized into three sections. First, the motivation of the present study is given as described above. Next, we recall some essential definitions needed in our work. In Section 3, a new concept of dominating proximal generalized Geraghty for two mappings is introduced. Then, we show that a common best proximity coincidence point of these mappings uniquely exists under some additional assumptions. Moreover, an example is provided to support the main result. Lastly, we consider some further results following from our main theorem.
In this section, we review some notations and important definitions to be used in the next section. Let be a pair of nonempty subsets of a metric space . We adopt the following notations:
Definition 1 (see [1, 26, 29]). Let and be mappings.
An element is said to be(i)A best proximity point of if(ii)A best proximity coincidence point of the pair if(iii)A common best proximity coincidence point of the pair if
Definition 2 (see ). Let be mappings. A pair is said to commute proximally if for each ,
3. Main Results
In this section, we introduce a class of pairs of some proximal generalized Geraghty contractions with dominating property and prove common best proximity theorem for this class.
Definition 3. Let be mappings. A pair is said to be dominating proximal generalized Geraghty if there exists such that for each ,implieswhere
Theorem 4. Let be a pair of nonempty subsets of a complete metric space , and let be mappings. Suppose that the pair is dominating proximal generalized Geraghty. Assume that and are nonempty such that is closed. If the following assertions hold:(i) and (ii) and are continuous(iii) and commute proximallythen there is only one common best proximity coincidence point of in .
Proof. Let be a fixed element in . From the assumption that , we get that for each element , there is an element such that . Then, we obtain a sequence in satisfyingfor each . Since , there exists an element such thatfor all Further, we obtain thatfor all .
Our first goal is to show that for some .
In the case that for some , by (11) and (12), we get thatSince and commute proximally, , and so we are done.
Now, for the harder part, assume that for all . From (12), note thatfor all . Since is dominating proximal generalized Geraghty, we have thatConsider thatThis implies that for all .
Next, we prove that the sequence converges to .
Consider the following two cases.
Case 1. .
From (15), we have thatfor all . Therefore, is a nonincreasing sequence which is bounded below, and so it is convergent. To obtain that suppose on the contrary that . By (17), letting impliesSince , by the definition of ,which is a contradiction. Thus, we get that .
Case 2. . Similarly, by (15), we have thatSince for all , we get that , and hence, By the definition of , we also have that
Due to both cases, we obtain the desired limitNow, we claim that is a Cauchy sequence.
Suppose contradiction, that is, is not a Cauchy sequence. Then, there exists such that there are subsequences and of so that for all with , we obtainIn addition, we can choose the smallest satisfying (22) for all so thatBy using (22) and (23), we have thatSince , taking the limit as in (24) impliesConsider, by the triangular inequality, thatConsequently, .
In the same way, we get thatand so .
Thus,Since and satisfy equations (11) and (12), we obtain thatfor each . Since is dominating proximal generalized Geraghty,whereBy (21), we observe thatand, as a consequence,Hence, (25) implies that . Then, by (28) and (30), we obtain thatBy the property of , we obtain thata contradiction. Therefore, we can conclude that is a Cauchy sequence.
The essential observation is that is a Cauchy sequence in the closed subset of the complete metric space . Then, there exists such that . Consider, by (11) and (12), that . Since and commute proximally,for all . By the continuity of and ,
We are now in a position to show that a best proximity coincidence point of exists. Since , there exists such that
By the assumption that and commute proximally, . According to the assumption that , there exists such that
Next, we claim that . Suppose that , i.e., . We observe that
Since , we have . By the property of , . This contradicts the assumption that . Thus, , and hence
That is, the element is a common best proximity coincidence point of .
Finally, we have to show that the point is unique.
Let be a common best proximity coincidence point of . Then
Notice that . Since is dominating proximal generalized Geraghty, we obtain that
If , then , and so, by using the property of ,a contradiction. Thus, must be zero. As a result, . The proof is now completed.
Example 5. Let equipped with the metric given byLet and It is easy to see that . Define the mappings byfor all Notice that and are continuous. To show that the pair is dominating proximal generalized Geraghty, define the mapping byThen, . Let satisfyingObserve that they must have the following forms:where , and . To obtain the inequality (9), if , then we are done. Assume that . Then, are all distinct. As a consequence, . Thus, we have thatTherefore, the pair is dominating proximal generalized Geraghty.
Next, consider, by the definition of and , that and . Additionally,Now, it remains to show that and commute proximally. Let such thatConsequently, , where and . Thus,Thus, and commute proximally.
Finally, by Theorem 4, we can conclude that there is a unique common best proximity coincidence point of the pair . In fact, the point is the unique common best proximity coincidence point of .
As a consequence of our result, the following corollaries are given. Precisely, these are the special cases of Theorem 4 when for , and for , respectively.
Corollary 6. Let be a pair of nonempty subsets of a complete metric space . Assume that and are nonempty such that is closed. If are mappings such that the following assertions hold:(1) and (2) and are continuous(3) and commute proximally(4)There exists such that for each implieswhere , then there is a unique common best proximity coincidence point of the pair .
Corollary 7. Let be a pair of nonempty subsets of a complete metric space . Assume that and are nonempty such that is closed. If are mappings such that the following assertions hold:(1) and (2) and are continuous(3) and commute proximally(4)There exists such that for each implieswhere , then there is a unique common best proximity coincidence point of the pair .
In this work, we give an idea of dominating proximal generalized Geraghty for a pair of mappings and give the existence and uniqueness theorem for a common best proximity coincidence point of these pairs in a complete metric space with some extra assumptions. Further, we present an example of this result. Now, we pose the following open problem.
Open Problem 1. Can Theorem 4 be extended to the framework of complete metric spaces endowed with graphs?
No data were used to support this study.
Conflicts of Interest
The authors have no conflict of interests regarding the publication of this paper.
The author would like to give special thanks to Assistant Professor Phakdi Charoensawan for all of his useful comments and suggestions. This research was partially supported by Chiang Mai University.
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